It is possible to write the Gorini-Kossakowski-Lindblad-Sudarshan
(GKLS) equation \(\dot{\rho } = \mathcal {L}[\rho ]\)  as an ordinary
matrix equation by introducing the orthonormal basis \(\lbrace  e_k\rbrace _{1 \le k \le D^2}\)  of matrices acting on \(\mathbb {C}^D\)[1]}, [2]}.
Then a density matrix \(\rho = \sum _k \rho _k e_k\)  is vectorized
as \(|\rho {\rangle }{\rangle }= \lbrace \rho _k\rbrace _{1 \le k \le D^2}\)  and
the superoperator \(\mathcal {L}\)  is represented as
a \(D^2 \times D^2\)  supermatrix \(L(t)_{jk}= \mathrm {Tr}(e_j^{\dagger } \mathcal {L}[e_k])\) , with which
the GKLS equation takes the form \(|\dot{\rho }{\rangle }{\rangle }= L(t)|\rho {\rangle }{\rangle }\) .
\(L(t)\)  can be put in the Jordan canonical form by a similarity
transformation \(C(t)\)  as \(L_J(t) = C^{-1}(t) L(t) C(t) =\mathrm {diag}(J_1(t), \ldots , J_N(t))\) .
The time-evolution may be defined as adiabatic if
there are no transitions among the Jordan blocks \(\lbrace J_s\rbrace \) . The similarity transformation \(C(t)\)  introduces
inter-block couplings through \(\dot{C}(t)\) , which may
be cancelled by the CD term that suppresses
the nonadiabatic transitions.
